New Short Cut Found For Long Math Proofs

By GINA KOLATA

Published: April 7, 1992

Correction Appended

IN a discovery that overturns centuries of mathematical tradition, a group of graduate students and young researchers has discovered a way to check even the longest and most complicated proof by scrutinizing it in just a few spots.

The finding, which some mathematicians say seems almost magical, depends upon transforming the set of logical statements that constitute a proof into a special mathematical form in which any error is so amplified as to be easily detectable.

Using this new result, the researchers have already made a landmark discovery in computer science. They showed that it is impossible to compute even approximate solutions for a large group of practical problems that have long foiled researchers. Even that negative finding is very significant, experts say, because in mathematics, a negative result, showing something is impossible, can be just as important and open just as many new areas of research as a positive one.

The discovery was made by Sanjeev Arora and Madhu Sudan, graduate students at the University of California at Berkeley, Dr. Rajeev Motwani, an assistant professor at Stanford University, and Dr. Carsten Lund and Dr. Mario Szegedy, young computer scientists at A.T.&T. Bell Laboratories. Dr. Motwani, who is the senior member of the group, just turned 30 on March 26.

"With the conventional notion of a proof, you had to check it line by line," said Dr. Michael Sipser, a theoretical computer scientist at the Massachusetts Institute of Technology. "An error might be buried on page 475, line 6. A 'less than or equal to' should have been a 'less than.' That would totally trash the whole proof. But you'd have to dig through the whole thing to find it," Dr. Sipser said. Now, he added, "the new idea is that there is a way to transform any proof so that if there is an error, it appears almost everywhere. I'd say, 'You have a proof? Show me a page.' If there is an error, it will be there."

The finding, which is built on two and a one half years work by leading researchers, is expected to have a profound impact. But because it is so new and unexpected, mathematicians and computer scientists cannot yet predict its scope of application.

Dr. Manuel Blum, a mathematician at the University of California at Berkeley, agreed that the result was "really exciting," with ramifications that were hard to predict. The new verification method can show up errors in calculations as well as in proofs, experts said, and Dr. Blum suggested it could have a role in checking long computer computations.

Dr. Umesh Vazirani, a theoretical computer scientist at the University of California at Berkeley, said that the discovery was "one of the most outstanding in the past decade," because it allowed investigators to decide, almost at a glance, whether it was worthwhile to try to find an approximate solution to a problem they were unable to solve exactly. "As soon as you formulate a problem, you can know if it is intractable," Dr. Vazirani said.

The finding is so recent that it has not yet been published. "We just wrote it in the last month or so," Dr. Motwani said. As is the custom in mathematics and theoretical computer science, the result gains credibility not by having been sent to a journal and reviewed by other scientists but by having been vouched for by leaders in the field. In this case, at least a dozen experts say they are convinced by the result, and amazed by it. High Praise for Discovery

"It is absolutely stunning," said Dr. Laszlo Babai, a theoretical computer scientist at the University of Chicago whose work helped lay the groundwork for the new discovery.

Dr. Richard Karp, a theoretical computer scientist at the University of California at Berkeley agreed. "Personally, to me it's very surprising," he said. Dr. Karp added that the method's application to show that certain problems have no approximate solutions is the most important discovery in his field of theoretical computer science in more than two decades.

The insight into the nature of proofs shows how a connection can be created between each and every logical statement of a proof, no matter how long or complicated the proof may be. It applies to the 15,000-page proof that is the longest ever published as well as to the single sentence proof that can be grasped in an instant. And it is no harder to check a long proof than a short one. "Even if your proof is the size of the universe, you still would not have an increased number of places to check," Dr. Babai said.

The investigators used a variety of mathematical tricks to transform a proof so that its errors, if any, will show up almost everywhere. One method they used was a way of comparing lists of data.

Data comparisons are an integral part of long proofs. For example, Dr. Babai pointed out, a typical mathematical proof might have a line that says, "We know by theorem 31 that a=b and by theorem 72 that b=c. Therefore, we conclude that a=c." To see if this statement is correct, Dr. Babai said, "We have to go back to theorem 31 and see that it does indeed say b=c, and similarly we have to look up theorem 72 and see that it has been copied correctly."

Correction: April 9, 1992, Thursday A diagram in Science Times on Tuesday about speedy checking of complex mathematical proofs incorrectly described a method that could be used to compare two lists of one million numbers each. The numbers would be arranged in mathematical constructs called 20-dimensional arrays, not in 20-by-20 arrays.